Problem: You have found the following ages (in years) of 5 sloths. The sloths are randomly selected from the 38 sloths at your local zoo: $ 16,\enspace 11,\enspace 10,\enspace 2,\enspace 10$ Based on your sample, what is the average age of the sloths? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 38 sloths, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $5$ samples and divide by $5$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{38.44} + {1.44} + {0.04} + {60.84} + {0.04}} {{5 - 1}} $ $ {s^2} = \dfrac{{100.8}}{{4}} = {25.2\text{ years}^2} $ We can estimate that the average sloth at the zoo is 9.8 years old. There is a variance of 25.2 years $^2$.